Remember (from the prologue): the hotter the water, the more compounds will be dissolved, and at a faster rate. Increasing the water temperature increases the temperature of the compounds themselves, which in turn increases their solubility.
In this video, Matt talks temperatures and the Arrhenius Equation
Temperature Equilibrium Calculator
This average brew water temperature calculator will help you consider the implications of changing your brew method and the change to brew water temperature that follows from that change. This will help you predict the difference in starting temperature needed to hit the same average brew water temperature.
Many brew methods get hotter as they process such as espresso, capsule, syphon and cezve coffee. A cupping or plunger, on the other hand, gets cooler after the water is added. In order to compare these brew methods, try this calculator. This calculator will be of particular use when trying to predict changes you may need to make from a grinder when it warms up. The temperature of the grinds will be much higher in peak periods compared to quiet times.
In this screen grab, Matt talks you through some real-world situations where you can use this calculator to predict what brew water temp changes may be needed to mitigate equipment heating up in busy periods.
As with refractometry, this sort of analysis should be paired with sensory analysis. One general trend is those brew methods that are hottest at the end of the brew cycle are likely to have a smaller window of success at the end of the brew cycle. Diffusion is the rate-limiting factor in extraction and diffusion rates are faster at higher temperatures.